Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.5 SURFACE INTEGRALS

Find the vector area of the surface of the hemispherex^2 +y^2 +z^2 =a^2 ,z≥ 0 ,by

evaluating the line integralS=^12

∮

Cr×draround its perimeter.

The perimeterCof the hemisphere is the circlex^2 +y^2 =a^2 , on which we have

r=acosφi+asinφj,dr=−asinφdφi+acosφdφj.

Therefore the cross productr×dris given by

r×dr=

∣

∣

∣∣

∣

∣

ijk

acosφasinφ 0

−asinφdφ acosφdφ 0

∣

∣

∣∣

∣

∣

=a^2 (cos^2 φ+sin^2 φ)dφk=a^2 dφk,

and the vector area becomes

S=^12 a^2 k

∫ 2 π

0

dφ=πa^2 k.

11.5.3 Physical examples of surface integrals

There are many examples of surface integrals in the physical sciences. Surface

integrals of the form (11.8) occur in computing the total electric charge on a

surface or the mass of a shell,

∫
Sρ(r)dS, given the charge or mass densityρ(r).
For surface integrals involving vectors, the second form in (11.9) is the most

common. For a vector fielda, the surface integral

∫
Sa·dSis called theflux
ofathroughS. Examples of physically important flux integrals are numerous.

For example, let us consider a surfaceSin a fluid with densityρ(r) that has a

velocity fieldv(r). The mass of fluid crossing an element of surface areadSin

timedtisdM=ρv·dSdt. Therefore thenettotal mass flux of fluid crossingS

isM=

∫
Sρ(r)v(r)·dS. As a another example, the electromagnetic flux of energy
out of a given volumeVbounded by a surfaceSis

∮

S(E×H)·dS.

The solid angle, to be defined below, subtended at a pointOby a surface (closed

or otherwise) can also be represented by an integral of this form, although it is

not strictly a flux integral (unless we imagine isotropic rays radiating fromO).

The integral

Ω=

∫

S

r·dS

r^3

=

∫

S

ˆr·dS

r^2

, (11.11)

gives thesolid angleΩsubtended atOby a surfaceSifris the position vector

measured fromOof an element of the surface. A little thought will show that

(11.11) takes account of all three relevant factors: the size of the element of

surface, its inclination to the line joining the element toOand the distance from

O. Such a general expression is often useful for computing solid angles when the

three-dimensional geometry is complicated. Note that (11.11) remains valid when

the surfaceSis not convex and when a single ray fromOin certain directions

would cutSin more than one place (but we exclude multiply connected regions).